CHEM 312 Lecture 9: Nuclear Reactions

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1 CHEM 312 Lecture 9: Nuclear Reactions Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and Radiochemistry, Chapter 4 Notation Energetics of Nuclear Reactions Reaction Types and Mechanisms Barriers Scattering Nuclear Reaction Cross Sections Reaction Observables Scattering Direct Reactions Compound Nuclear Reactions Photonuclear Reactions Nucleosynthesis 9-1

2 Nuclear Reactions Nucleus reactions with a range of particles nucleus, subatomic particle, or photon to produce other nuclei Short time frame (picosecond) First nuclear reaction from Rutherford What reaction was this? Number of terms conserved during nuclear reactions Number of nucleons except in reactions involving creation or annihilation of antinucleons charge Energy momentum angular momentum parity Q is the energy of the reaction 1.193MeV atom positive Q corresponds to energy release negative Q to energy absorption Q terms given per nucleus transformed N + He O+ H + Q 2 N(α, p) 17 8 O Q= MeV 6.02E23atoms 1.6E 13J x x mole MeV 1 = 1.15E11 J mole 9-2

3 Energetics Reaction Q values Not necessarily equal to kinetic energy of bombarding particles for the reaction to occur Need more energy than Q value for reaction to occur * Reaction products will have kinetic energy that needs to come from reaction Conservation of momentum Some particles kinetic energy must be retained by products as kinetic energy Amount retained as kinetic energy of products Based on projectile mass Retained kinetic energy becomes smaller with increasing target mass Equation for kinetic energy (T): APr ojectile T = Q What does this mean about reaction APr ojectile + AT arg et Heavier target or heavier projectile? 248 Cm + 18 O 266 Rf 248 T Q = Q = 248 Cm Projectile 18 T = Q = Q O Projectile 9-3

4 Energetics: Reaction Barrier Need to consider comparison of laboratory and center of mass frame Laboratory frame conservation of momentum considers angle of particles m m x p 2 Q = Tx ( 1+ ) Tp (1 + ) ( mptpmxtx ) cosθ mr mr mr Center of mass Total particle angular momentum is zero T cm 2 ( mp + mt ) v p p cm = vcm = 2 ( mp + mt ) Kinetic energy carried by projectile (T lab ) is not fully available for reaction T lab - T cm = T 0 For reaction to occur Q + T 0 must be achieved Basis for threshold reaction Q + T 0 > 0 v m T Q = T cm = T x lab T ( m p p T m + p m R T ) 9-4

5 Reaction Barrier Threshold energy (minimum energy for reaction) mp Q + Tlab TCM 0; Tcm = Tlab( ) Solve of laboratory T m + m T T T lab lab lab T lab ( m (1 ( m p p mp + m mp + m p T T Q mp (1 ( m + m ) Q )) Q T )) = m ( m p p p + m + m T T Q ( m mp + m Q mt m + m APr ojectile + AT arg et T Q MeV AT arg et Fraction of bombarding particle s kinetic energy retained as kinetic energy of products becomes smaller with increasing mass of target Heavier target or heavier projectile? 248 Cm + 18 O 266 Rf T p T )) = p T A for mass 9-5

6 Reaction Barrier: Threshold Energy Consider the 14 N(α,p) 17 O reaction Find threshold energy APr ojectile + AT arg et T Q MeV Q from mass excess AT arg et * Q= (-0.809) = MeV T ( )1.19 MeV = 1. 53MeV 14 Reaction barrier also induced by Coulomb interaction Need to have enough energy to react and overcome Coulomb barrier From charge repulse as particle approach each other * R is radius 2 Z1Z2e 1/ 3 * r o =1.1 to 1.6 fm Vc = R = ro A R Equation can vary due to r 1 + R2 o Z1Z2 V c can be above threshold energy V = 2*7 c = 0.96 MeV = 3. 36MeV 1 / 3 1/ V Center of mass, need to bring to laboratory frame Consider kinetic energy carried by projectile 3.36x ((14+4)/14) = 4.32 MeV alpha needed for reaction c 1/ 3 1/ 3 A1 + A2 MeV 9-6

7 Cross Section Limits Reaction cross section of πr 2 is approximated at high energies Wave nature of incident particle causes upper limit of reaction cross section to include de Broglie wavelength So cross section can be larger that area σ = π r ( R + ) Collision between neutron and target nucleus characterized by distance of closest approach B is impact parameter 2 9-7

8 σ = π [( l + 1) l ] = π (2l l Cross section + 1) Quantum-mechanical treatment T is the transmission coefficient for reaction of a neutron with angular momentum Represents fraction of incident particles with angular momentum that penetrate within range of nuclear forces Provides summing term to increase cross section Reason why cross section can be larger than physical size of nucleus σ l is partial cross section of given angular momentum l σ ( 2l + 1) 2 r = π T l l= 0 General trends for neutron and charged particles Charged particle cross section minimal at low energy Neutron capture cross section maximum at low energy 9-8

Types of Experiments: Excitation Functions variation of reaction cross section with incident energy shape can be determined by exposing several target foils in same beam with energy-degrading provide

9 Types of Experiments: Excitation Functions variation of reaction cross section with incident energy shape can be determined by exposing several target foils in same beam with energy-degrading provide information about probabilities for emission of various kinds of particles and combinations of particles in nuclear reactions formation of given product implies what particles were ejected from the target nuclide Range of cross sections can be evaluated 9-9

Low-Energy Reactions with Light Elastic scattering kinetic energy conserved Slow-Neutron Reactions Purest example of compoundnucleus behavior 1/v law governs most neutron cross sections in region of

10 Low-Energy Reactions with Light Elastic scattering kinetic energy conserved Slow-Neutron Reactions Purest example of compoundnucleus behavior 1/v law governs most neutron cross sections in region of thermal energies neutrons available only from nuclear reactions Range of energies can be obtained Deuteron Reactions Prevalence of one nucleon stripping large size and loose binding of deuteron Only proton and neutron in deuteron nucleus * Proton charge carries both nucleons Projectiles 9-10

11 High Energy Reactions Spallation Products products in immediate neighborhood of target element found in highest yields within 10 to 20 mass numbers yields tend to form in two regions β stability for medium-weight products neutron-deficient side of stability with increasing Z of products Used to produce beam of neutrons at spallation neutron source Heavy Z will produce neutrons Basis of Spallation neutron source ( High-Energy Fission single broad peak in mass-yield curve instead of double hump seen in thermal-neutron fission many neutron-deficient nuclides especially among heavy products originate from processes involving higher deposition energies lower kinetic energies do not appear to have partners of comparable mass arise from spallation-like or fragmentation reactions 9-11

12 High-Energy Reactions Mass-Yield Curves at low energies, compound-nucleus picture dominates as energy increases importance of direct reactions and preequilibrium (pre-compound nucleus) emission increase above 100 MeV, nuclear reactions proceed nearly completely by direct interactions products down to mass number 150 are spallation products those between mass numbers 60 and 140 are fission products Cascade-Evaporation Model Above 100 MeV reactions energy of the incident proton larger than interaction energy between the nucleons in the nucleus Wavelength less than average distance between nucleons proton will collide with one nucleon at a time within the nucleus * high-energy proton makes only a few collisions in nucleus * Produces nucleons with high energy 9-12

13 Heavy Ion Reactions Inelastic scattering scattering in which some of projectile s kinetic energy transformed into excitation of target nucleus greatest importance at large impact parameters heavy ions valuable can excite high-spin states in target nuclei because of large angular momenta Can experience Coulomb excitation high charges below Coulomb barrier heights and excite nuclei by purely electromagnetic interactions Transfer Reactions stripping and pickup reactions prevalent with heavy ions take place at impact parameters just below those at which interactions are purely Coulombic angular distributions show oscillatory, diffraction-like pattern when transfer reaction to single, well-defined state observed 9-13

14 Heavy Ion Reactions: Deep Inelastic Reactions Relatively large amounts of nuclear matter transferred between target and projectile Show strongly forward-peaked angular distributions Grazing contact mechanism Products with masses in vicinity of projectile mass appear at angles other than classical grazing angle Relatively small kinetic energies Total kinetic energies of products strongly correlated with amount of mass transfer Increasing mass difference of product and projectile lowers kinetic energy Product will dissociate into two fragments Appreciable fraction of incident kinetic energy dissipated and goes into internal excitation 9-14

15 Compound-Nucleus Reactions compound-nucleus formation can only take place over a restricted range of small impact parameters can define critical angular momentum above which complete fusion cannot occur σ cf /σ R decreases with increasing bombarding energy Neutron deficient heavy ions produce compound nuclei on neutron-deficient side of β stability belt Heavy ion of energy above Coulomb barrier brings enough excitation energy to evaporate several nucleons 5-10 MeV deexcitation for neutron evaporation heavy-ion reactions needed for reaching predicted island of stability around Z=114 to Z=184 U is excitation energy, M A and M a masses of target and projectile, T a is projectile kinetic energy, S a is projectile binding energy in compound nucleus M A U = Ta + M A + M a S a 9-15

16 Photonuclear reactions Reactions between nuclei and lowand medium-energy photons dominated by giant resonance Excitation function for photon absorption goes through a broad maximum a few MeV wide Due to excitation of dipole vibrations of protons against neutrons in the nucleus Resonance peak varies smoothly with A 24 MeV at 16 O 13 MeV at 209 Bi Peak cross sections are mb (γ, p), (γ, n), (γ,α) reactions /cuos/researchgroups/hfs/research /photonuclear_reactions.html 9-16

17 Origin of Elements Gravitational coalescence of H and He into clouds Increase in temperature to fusion Proton reaction 1 H + n 2 H + γ 2 H + 1 H 3 He 2 H + n 3 H 3 H + 1 H 4 He + γ 3 He + n 4 He + γ 3 H + 2 H 4 He + n 2 H + 2 H 4 He + γ 4 He + 3 H 7 Li + γ 3 He+ 4 He 7 Be + γ 7 Be short lived Initial nucleosynthesis lasted 30 minutes * Consider neutron reaction and free neutron half life Further nucleosynthesis in stars No EC process in stars 9-17

Stellar Nucleosynthesis He burning 4 He + 4 He 8 Be + γ - 91.

367 MeV 12 C + 4 He 16 O 16 O + 4 He 20 Ne Formation of 12 C based on

18 Stellar Nucleosynthesis He burning 4 He + 4 He 8 Be + γ kev Too short lived 3 4 He 12 C + γ MeV 12 C + 4 He 16 O 16 O + 4 He 20 Ne Formation of 12 C based on Hoyle state Excited nuclear state Somewhat different from ground state 12 C Around 7.6 MeV above ground state

19 Stellar Nucleosynthesis CNO cycle 12 C + 1 H 13 N + γ 13 N 13 C + e + + νe 13 C + 1 H 14 N + γ 14 N + 1 H 15 O + γ 15 O 15 N + e + + νe 15 N + 1 H 12 C + 4 He Net result is conversion of 4 protons to alpha particle 4 1 H 4 He +2 e νe +3 γ Fusion up to Fe Binding energy curve 9-19

20 Formation of elements A>60 Neutron Capture; S-process A>60 68 Zn(n, γ) 69 Zn, 69 Zn 69 Ga + β + ν mean times of neutron capture reactions longer than beta decay half-life Isotope can beta decay before another capture Up to Bi 9-20

21 Nucleosynthesis: R process Neutron capture time scale very much less than β- decay lifetimes Neutron density /m 3 Extremely high flux capture times of the order of fractions of a second Unstable neutron rich nuclei rapidly decay to form stable neutron rich nuclei all A<209 and peaks at N=50,82, 126 (magic numbers) 9-21

22 P process Formation of proton rich nuclei Proton capture process 70<A<200 Photonuclear process, at higher Z (around 40) (γ, p), (γ,α), (γ, n) 190 Pt and 168 Yb from p process Also associated with proton capture process (p,γ) Variation on description in the literature 9-22

23 Proton-rich nuclei with Z = 7-26 (p,γ) and β + decays that populate the p- rich nuclei Also associated with rapid proton capture process Initiates as a side chain of the CNO cycle 21 Na and 19 Ne Forms a small number of nuclei with A< 100 rp process (rapid proton capture) 9-23

24 Review Notes Understand Reaction Notation Understand Energetics of Nuclear Reactions Q values and barriers Understand the Different Reaction Types and Mechanisms Particles Energy Relate cross sections to energy Describe Photonuclear Reactions Routes and reactions in nucleosynthesis Influence of reaction rate and particles on nucleosynthesis 9-24

25 Questions Describe the different types of nuclear reactions shown on Provide notations for the following Reaction of 16 O with 208 Pb to make stable Au Formation of Pu from Th and a projectile Find the threshold energy for the reaction of 59 Co and an alpha that produces a neutron and a product nuclei What are the differences between low and high energy reactions? How does a charged particle reaction change with energy? A neutron reaction? How are actinides made in nucleosynthesis? What is the s-process? What elements were produced in the big bang? Which isotopes are produced by photonuclear reactions? What is interesting about the production of 12 C 9-25

26 Pop Quiz Provide the Q value, threshold energy, and Coulomb barrier for the compound nucleus reaction of 18 O with 244 Cm Provide comment in blog Bring to next class (31 October) 9-26

27 CHEM 312 Lecture 10: Chemical Speciation Use of constants to model chemical form Thermodynamic and kinetic Determine property of radioelement based on speciation Chemical species in system Review Equilibrium constants Activity Use of constants in equation 9-27

28 Reaction Constants For a reaction aa + bb <--> cc + dd At equilibrium ratio of product to reactants is a constant Constant can change with conditions Not particularly constant By convention, constants are expressed as products over reactants K = [C]c [D] d [A] a [B] b Conditions under which the constant is measured should be listed Temperature, ionic strength 9-28

29 Complete picture: Activities Strictly speaking, activities, not concentrations should be used c d γ C[ C] γ D[ D] K = a b γ A[ A] γ B[ B] Activities normalize concentration to amount of anions and cations in solution At low concentration, activities are assumed to be 1 constant can be evaluated at a number of ionic strengths and overall activities fit to equations Debye-Hückel (Physik Z., 24, 185 (1923)) log γ A = Z a 2 µ R A µ Z A = charge of species A µ = molal ionic strength R A = hydrated ionic radius in Å (from 3 to 11) First estimation of activity 9-29

30 Activities Debye-Hückel term can be written as: D = µ µ Specific ion interaction theory Uses and extends Debye-Hückel long range Debye-Hückel Short range ion interaction term ε ij = specific ion interaction term log ß(µ) = log ß(0) + Z i 2 D εij µ Pitzer Binary (3) and Ternary (2) interaction parameters log γ i = Z 2 D + ε ij µ 9-30

Experimental Data shows change in stability constant with ionic strength 6.6 6.

31 Experimental Data shows change in stability constant with ionic strength Ion Specific Interaction Theory Cm-Humate at ph logß K Ca 2+ sqrt Im Al 3+ Fe(CN)

32 Constants Constants can be listed by different names Equilibrium constants (K) Reactions involving bond breaking * 2 HX <--> 2H + + X 2 2- Stability constants (ß), Formation constants (K) Metal-ligand complexation * Pu 4+ + CO 3 2- <--> PuCO 3 2+ * Ligand is written in deprotonated form Conditional Constants An experimental condition is written into equation * Pu 4+ + H 2 CO 3 <--> PuCO H + Constant can vary with concentration, ph Must look at equation! 9-32

33 Using Equilibrium Constants Constants and balanced equation can be used to evaluate concentrations at equilibrium [ H ] [ X 2 HX <--> 2H + 2 ] + X 2- K = 2 2 [ HX ] K=4E-15 If you have one mole of HX initially, what are the concentration of all species at equilibrium? Try to write species in terms of one unknown Start with species of lowest concentration [X 2 2- ]=x, [H + ]=2x, [HX]=1-2x, Since K is small, x must be small Use the approximation 1-2x 1 Substitute x and rearrange K Solve for x [X 2 2- ]=1E-5, [H + ]=2E [ x][2x] [ x][2x] K = = = 2 [1 2x] 1 4E 15 = 1E 15 = x x = 1E 5 4x 3 3 4x

34 Realistic Case Metal ion of interest may be in complicated environment May different species to consider simultaneously Consider uranium in an aquifer Example is still a simplified case Species to consider in this example include free metal ion: UO 2 2+ hydroxides: (UO 2 ) x (OH) y carbonates: UO 2 CO 3 humates: UO 2 HA(II), UO 2 OHHA(I) Need to get stability constants for all species Example: UO CO 3 2- <--> UO 2 CO 3 Know or find conditions Total uranium, total carbonate, ph, total humic concentration 9-34

35 Stability constants for selected uranium species at 0.1 M ionic strength Species logß UO 2 OH UO 2 (OH) UO 2 (OH) UO 2 (OH) (UO 2 ) 2 OH (UO 2 ) 2 (OH) UO 2 CO UO 2 (CO 3 ) UO 2 (CO 3 ) UO 2 HA(II) 6.16 UO 2 (OH)HA(I) 14.7±0.5 Other species may need to be considered. If total uranium concentration is low enough, binary or tertiary species can be excluded. Chemical thermodynamics of uranium:

36 Equations Write concentrations in terms of species Total uranium in solution, [U] tot, is the sum of all solution phase uranium species [U] tot = UO 2 2+ free +U-carb+U-hydroxide+U-humate [CO 3 2- ] free =f(ph) From Henry s constant for CO 2 and K 1 and K 2 from CO 3 H 2 log[co 3 2- ] free =logk H K 1 K 2 +log(pco 2 )-2log[H + ] * With -log[h + ]=ph log[co 3 2- ] free =logk H K 1 K 2 +log(pco 2 )+2pH [OH - ] = f(ph) [HA] tot = UO 2 HA + UO 2 OHHA+ HA free 9-36

37 Uranium speciation equations Write the species in terms of metal, ligands, and constants Generalized equation, with free uranium, free ligand A and free ligand B [( UO2) x Aa Bb ] 2+ x a b β xab = 2+ x a b [( UO 2) x Aa Bb ] = β xab[ UO2 ] [ A] [ B] [ UO2 ] [ A] [ B] Provide free ligand and metal concentrations as px value px = -log[x] free puo 2+ 2 =-log[uo 2+ 2 ] Rearrange equation with px values Include logβ xab, treat as px term [(UO 2 ) x A a B b ] = 10 -(xpuo2+apa+bpb-log xab ) Specific example for (UO 2 ) 2 (OH) 2+ 2 [(UO 2 ) 2 (OH) 2+ 2 ]=10 -(2pUO2+2pOH-22.0) Set up equations where total solution uranium concentration is sum of all species and solve for known terms 9-37

38 Speciation calculations: Excel spreadsheets CHESS Program 9-38

39 U speciation with different CO 2 partial pressure Mole Fraction of U(VI) Species UO % CO 2 1% CO 2 UO 2 HA(II) - UO 2 (OH) 3 UO 2 OHHA(I) UO 2 (OH) 2 Mole Fraction of U(VI) Species UO 2 2+ UO 2 HA(II) UO OHHA(I) 2 UO 2 (CO 3 ) 3 4- UO 2 (CO 3 ) ph ph 1.0 Mole Fraction of U(VI) Species UO 2 2+ UO 2 HA(II) UO 2 OHHA(I) UO 2 (CO 3 ) 3 4- UO 2 (CO 3 ) % CO ph 9-39

40 Comparison of measured and calculated uranyl organic colloid 1.0 [U-colloid] [U(VI)] total % 100% 1% 0% % ph 9-40

41 Energy terms Constants can be used to evaluate energetic of reaction From Nernst equation G=-RTlnK G= H-T S -RTlnK = H-T S RlnK= - H/T + S * Plot RlnK vs 1/T Rlnß Temperature effect on Np-Humate stability Temp ( C) /T (K) ²H = ± 2.8 kj/mol ²G 298 =-21.7 kj/mol ²S=1.2±1.4 J/molK 9-41

42 Solubility Products Equilibrium involving a solid phase AgCl(s) <--> Ag + + Cl - K = [Cl ][Ag + ] [AgCl] AgCl concentration is constant Solid activity and concentration is treated as constant By convention, reaction goes from solid to ionic phase in solution Can use K sp for calculating concentrations in solution K sp = K[AgCl] = [Cl ][Ag + ] 9-42

43 Solubility calculations AgCl(s) at equilibrium with water at 25 C gives 1E-5 M silver ion in solution. What is the K sp?? AgCl(s) <--> Ag + + Cl - : [Ag + ] = [Cl - ] K sp = 1E-5 2 = 1E-10 What is the [Mg 2+ ] from Mg(OH) 2 at ph 10? K sp = 1.2E-11= [Mg 2+ ] [OH] 2 [OH] = 10 -(14-10) [Mg 2+ ] = 1.2E 11 1E 8 = 1.2E

44 Solubility calculations K sp of UO 2 = What is the expected U 4+ concentration at ph 6. Generalize equation for any ph Solubility reaction: UO H 2 O U(OH) 4 U OH - K sp = [U 4+ ][OH - ] 4 [U 4+ ]= K sp /[OH - ] 4 poh + ph =14 At ph 6, poh = 8, [OH - ]=10-8 [U 4+ ]= /[10-8 ] 4 = /10-32 = M For any ph [U 4+ ]= /[10 -(14-pH)*4 ] Log [U 4+ ]= -52+((14-pH)*4) 9-44

45 Limitations of K sp Solid phase formation limited by concentration below 1E-5/mL no visible precipitate forms colloids formation of supersaturated solutions slow kinetics Competitive reactions may lower free ion concentration Large excess of ligand may form soluble species AgCl(s) + Cl - <--> AgCl 2- (aq) K sp really best for slightly soluble salts 9-45

CHEM 312: Lecture 9 Part 1 Nuclear Reactions

CHEM 312: Lecture 9 Part 1 Nuclear Reactions Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and Radiochemistry, Chapter 4 Notation Energetics of Nuclear Reactions Reaction Types and Mechanisms